Ray-to-projector is well-defined and bijective

Let $u$ be unit and suppose $v=\alpha u$ is another unit representative of the same ray. Then $1=\mathrm{\parallel }v{\mathrm{\parallel }}^2=\mathrm{\mid }\alpha {\mathrm{\mid }}^2\mathrm{\parallel }u{\mathrm{\parallel }}^2$, so $\mathrm{\mid }\alpha \mathrm{\mid }=1$. By [F2], $P_{v}z=⟨z,\alpha u⟩\alpha u=\bar{\alpha }\alpha ⟨z,u⟩u=P_{u}z$ for every $z$. Thus the projector does not depend on the unit representative.

By [F2], $P_u^{2}z=⟨z,u⟩⟨u,u⟩u=P_{u}z$. Its range is $\mathbb{C}u$, a one-dimensional subspace, so [F7] gives rank one. For every $z$, $P_{u}z\in \mathbb{C}u$ and $⟨z-P_{u}z,u⟩=⟨z,u⟩-⟨z,u⟩⟨u,u⟩=0,$ so the residual is in $(\mathbb{C}u{)}^{\perp }$; by the residual characterization in [F3], $P_u$ is the orthogonal projection onto $\mathbb{C}u$. Also, for all $z,w$, $⟨P_{u}z,w⟩=⟨z,u⟩⟨u,w⟩=⟨z,P_{u}w⟩$ and $⟨P_{u}z,z⟩=\mathrm{\mid }⟨z,u⟩{\mathrm{\mid }}^2\ge 0$. Hence $P_u$ is non-negative by [F4]. Choose an orthonormal basis $(e_j)$ by [F6] and write $u={\sum }_j{c}_j{e}_j$; then $⟨P_u{e}_j,e_j⟩=\mathrm{\mid }{c}_j{\mathrm{\mid }}^2$, so [F5] gives $\mathrm{tr}(P_u)={\sum }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2=⟨u,u⟩=1$ by orthonormality.

Conversely, let $P$ be a rank-one orthogonal projection onto $W$. By [F7], $\dim W=1$, so choose a unit $u\in W$. For every $z$, write $Pz=\alpha u$; [F3] gives $z-Pz\in W^{\perp }$, so $⟨z,u⟩=⟨Pz,u⟩=\alpha$. Therefore $Pz=⟨z,u⟩u=P_{u}z$, proving surjectivity. If $P_u=P_v$, their ranges are equal, so $u$ and $v$ span the same line and represent the same ray. This proves injectivity and the claimed bijection.